Interior pointwise Cα regularity for elliptic and parabolic equations with divergence-free drifts

We investigate the interior pointwise Cα regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior Cα regularity for some 0<α<1 has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise Cα regularity for any 0<α<1 provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.